General Balanced Random Effects Model

  • Hardeo Sahai
  • Mario Miguel Ojeda


In previous chapters, we have considered random effects models for various crossed and nested designs with equal numbers in all cells and subclasses, and which are called balanced complete models. In this chapter, we present a unified treatment of balanced random effects models in terms of the so-called general linear model.


Variance Component Unbiased Estimator Confidence Coefficient Exact Confidence Interval Approximate Confidence Interval 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. R. Ahmad and S. M. Mostafa (1987), Empirical Bayes estimation of variance components in balanced random models, J. Statist Comput. Simul, 27, 143–153.CrossRefGoogle Scholar
  2. A. Albert (1976), When is a sum of squares an analysis of variance?, Ann. Statist, 4, 775–778.MathSciNetCrossRefGoogle Scholar
  3. M. S. Bartlett (1936), The information available in small samples, Proc. Cambridge Philos. Soc, 32, 560–566.CrossRefGoogle Scholar
  4. M. S. Bartlett (1953), Approximate confidence intervals II: More than one unknown parameter, Biometrika, 40, 306–317.MathSciNetGoogle Scholar
  5. L. D. Broemeling (1969), Confidence intervals for variance ratios of random model, J. Amer. Statist Assoc., 64, 660–664.CrossRefGoogle Scholar
  6. K. A. Brownlee (1965), Statistical Theory and Methodology in Science and Engineering, 2nd ed., Wiley, New York.Google Scholar
  7. M. G. Bulmer (1957), Approximate confidence limits for components of variance, Biometrika, 44, 159–167.MathSciNetGoogle Scholar
  8. R. K. Burdick, and F. A. Graybill (1988), The present status of confidence interval estimation on variance components in balanced and unbalanced random models, Comm. Statist A Theory Methods, 17, 1165–1195.MathSciNetCrossRefGoogle Scholar
  9. R. K. Burdick and F. A. Graybill (1992), Confidence Intervals on Variance Components, Marcel Dekker, New York.Google Scholar
  10. R. K. Burdick and R. L. Sielken, Jr. (1978), Exact confidence intervals for linear combinations of variance components in nested classifications, J. Amer Statist Assoc., 73, 632–635.CrossRefGoogle Scholar
  11. G. Casella and R. L. Berger (2002), Statistical Inference, 2nd ed., Duxbury, Pacific Grove, CA.Google Scholar
  12. W. G. Cochran (1951), Testing a linear relation among variances, Biometrics, 7, 17–32.MathSciNetCrossRefGoogle Scholar
  13. P. Cook, L. D. Broemeling, and M. Gharaff (1990), A Bayesian analysis of the mixed linear model, Comm. Statist. A Theory Methods, 19, 987–1002.MathSciNetCrossRefGoogle Scholar
  14. R. Das and B. K. Sinha (1987), Robust optimum invariant unbiased tests for variance components, in T. Pukkila and S. Puntanen, eds., Proceedings of Second International Tampere Conference on Statistics, University of Tampere, Tampere, Finland, 317–342.Google Scholar
  15. J. M. Davenport (1975), Two methods of estimating degrees of freedom of an approximate F, Biometrika, 62, 682–684.MathSciNetCrossRefGoogle Scholar
  16. F. A. Graybill (1954), On quadratic estimates of variance components, Ann. Math. Statist, 25, 367–372.MathSciNetCrossRefGoogle Scholar
  17. F. A. Graybill (1961), An Introduction to Linear Statistical Models, Vol. I, McGraw-Hill, New York.Google Scholar
  18. F. A. Graybill (1976), Theory and Application of the Linear Model, Duxbury, North Scituate, MA.Google Scholar
  19. F. A. Graybill and R. A. Hultquist (1961), Theorems concerning Eisenhart’s Model II, Ann. Math. Statist, 32, 261–269.MathSciNetCrossRefGoogle Scholar
  20. F. A. Graybill and C.-M. Wang (1980), Confidence intervals on nonnegative linear combinations of variances, J. Amer. Statist. Assoc., 75, 869–873.MathSciNetCrossRefGoogle Scholar
  21. F. A. Graybill and A. W. Wortham (1956), A note on uniformly best unbiased estimators for variance components, J. Amer. Statist. Assoc., 51, 266–268.MathSciNetCrossRefGoogle Scholar
  22. H. O. Hartley (1967), Expectations, variances and covariances of ANOVA mean squares by synthesis, Biometrics, 23, 105–114; corrigenda, 23, 853.MathSciNetCrossRefGoogle Scholar
  23. J. Hartung and B. Voet (1986), Best invariant unbiased estimators for the mean squared error of variance components estimators, J. Amer. Statist. Assoc., 81, 689–691.MathSciNetCrossRefGoogle Scholar
  24. L. H. Herbach (1959), Properties of Model II type analysis of variance tests A: Optimum nature of the F-test for Model II in the balanced case, Ann. Math. Statist, 30, 939–959.MathSciNetGoogle Scholar
  25. R. B. Howe and R. H. Myers (1970), An alternative to Satterthwaite’s test involving positive linear combinations of variance components, J. Amer. Statist. Assoc., 65, 404–412.CrossRefGoogle Scholar
  26. A. Huitson (1955), A method of assigning confidence limits to linear combinations of variances, Biometrika, 42, 471–473.MathSciNetGoogle Scholar
  27. K. M. S. Humak (1984), Statistische Methoden dwe Modellbildung III: Statistische Inferenzfur Kovarianzparameter, Akademia-Verlag, Berlin.Google Scholar
  28. A. I. Khuri (1981), Simultaneous confidence intervals for functions of variance components in random models, J. Amer. Statist. Assoc., 76, 878–885.MathSciNetCrossRefGoogle Scholar
  29. A. W. Kimball (1951), On dependent tests of significance in the analysis of variance, Ann. Math. Statist., 22, 600–602.MathSciNetCrossRefGoogle Scholar
  30. E. L. Lehmann and H. Scheffé (1950), Completeness, similar regions and unbiased estimation, Sankhyā, 10, 305–340.Google Scholar
  31. T.-F. C. Lu (1985), Confidence Intervals on Sums, Differences and Ratios of Variances Components, Ph.D. dissertation, Colorado State University, Fort Collins, CO.Google Scholar
  32. T. Mathew and B. K. Sinha (1988), Optimum tests for fixed effects and variance components in balanced models, J. Amer. Statist. Assoc., 83, 133–135.MathSciNetCrossRefGoogle Scholar
  33. T. Mathew, B. K. Sinha, and B. C. Sutradhar (1992), Nonnegative estimation of variance components in general balanced mixed models, in A. Md. E. Saleh, ed., Nonparametric Statistics and Related Topics, Elsevier, New York, 281–295.Google Scholar
  34. R. H. Myers and R. B. Howe (1971), On alternative approximate F tests of hypotheses involving variance components, Biometrika, 58, 393–396.MathSciNetGoogle Scholar
  35. H. Raiffa and R. Schlaifer (1961), Applied Statistical Decision Theory, Harvard University Press, Cambridge, MA.Google Scholar
  36. P. Rudolph (1976), Bayesian estimation in the linear models under different assumptions about the covariance structure: Variance components models, equicorrelation model, Math. Oper. Statist. Ser. Statist., 7, 649–665.MathSciNetGoogle Scholar
  37. H. Sahai and R. L. Anderson (1973), Confidence regions for variance ratios of random models for balanced data, J. Amer. Statist. Assoc., 68, 951–952.MathSciNetCrossRefGoogle Scholar
  38. H. Scheffé (1943), On the solutions of the Behrens-Fisher problem, based on the t distribution, Ann. Math. Stat., 14, 35–44.CrossRefGoogle Scholar
  39. J. F. Seely (1980), Some remarks on exact confidence intervals for positive linear combinations of variance components, J. Amer. Statist. Assoc., 75, 372–374.MathSciNetCrossRefGoogle Scholar
  40. B. Seifert (1978), A note on the UMPU character of a test for the mean in balanced randomized nested classification, Statistics, 9, 185–189.MathSciNetGoogle Scholar
  41. B. Seifert (1979), Optimal testing for fixed effects in general balanced mixed classification models, Statistics, 10, 237–256.MathSciNetGoogle Scholar
  42. B. Seifert (1981), Explicit formulae of exact tests in mixed balanced ANOVA models, Biometrical J., 23, 535–550.MathSciNetCrossRefGoogle Scholar
  43. B. Seifert (1985), Estimation and tests of variance components using the MINQUE method, Statistics, 6, 621–635.MathSciNetGoogle Scholar
  44. A. M. Skene (1983), Computing marginal distributions for the dispersion parameters of analysis of variance models, Statistician, 32, 99–108.CrossRefGoogle Scholar
  45. E. Spjøtvoll (1967), Optimun invariant test in unbalanced variance components models, Ann. Math. Statist., 38, 422–429.MathSciNetCrossRefGoogle Scholar
  46. G. C. Tiao and I. Guttman (1965), The inverted Dirichlet distribution with applications, J. Amer. Statist. Assoc., 60, 793–805.MathSciNetCrossRefGoogle Scholar
  47. N. Ting, R. K. Burdick, and F. A. Graybill (1991), Confidence intervals on ratios of positive linear combinations of variance components, Statist. Probab. Lett., 11, 523–528.MathSciNetCrossRefGoogle Scholar
  48. N. Ting, R. K. Burdick, F. A. Graybill, and R. Gui (1989), One-sided confidence intervals on non-negative sums of variance components, Statist. Probab. Lett, 8, 129–135.MathSciNetCrossRefGoogle Scholar
  49. N. Ting, R. K. Burdick, F. A. Graybill, S. Jeyaratnam, and T.-F. C. Lu (1990), Confidence intervals on linear combinations of variance components that are unrestricted in sign, J. Statist. Comput. Simul, 35, 135–143.MathSciNetCrossRefGoogle Scholar
  50. C. M. Wang (1988), One-sided confidence intervals for the positive linear combination of two variances, Comm. Statist. B Simul. Comput, 17, 283–292.CrossRefGoogle Scholar
  51. C. M. Wang (1991), Approximate confidence intervals on positive linear combinations of expected mean squares, Comm. Statist. B Simul. Comput., 20, 81–96.CrossRefGoogle Scholar
  52. B. L. Welch (1956), On linear combinations of several variables, J. Amer. Statist. Assoc., 51, 132–148.MathSciNetCrossRefGoogle Scholar
  53. H. Yassaee (1981), On integrals of Dirichlet distributions and their applications, Comm. Statist. A Theory Methods, 10, 897–906.MathSciNetCrossRefGoogle Scholar
  54. L. Zhou and T. Mathew (1994), Some tests for variance components using generalized p-values, Technometrics, 36, 394–402.MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Hardeo Sahai
    • 1
  • Mario Miguel Ojeda
    • 2
  1. 1.Center for Addiction Studies School of MedicineUniversidad Central del CaribeBayamon, Puerto RicoUSA
  2. 2.Económico AdministrativaUniversidad VeracruzanaKalapa, VeracruzMéxico

Personalised recommendations